I am working in the book "A Hilbert space Problem book", written by Halmos.
There is a claim on page 109 stating that:
"On a finite dimensional space, every hyponormal operator is normal".
In which, "hyponormal operator" implies some operator $A$ with $A^*A \ge AA^*$.
In order to prove it, one assume that $A$ is hyponormal, then $\operatorname{tr} \left( A^*A- AA^* \right) = 0$.
They also claimed that every positive operator on finite dimensional vector space with trace $0$ is $0$ operator.
Could you please prove the last claim? I have no idea to verify it.